In the high-pressure fuel injection pipe having the connecting head of that kind in the prior art, as shown in FIG. 12 and FIG. 13, as a frustrum connecting head 12 having a seat face 13 formed of an outer circumference at the end portion of a thick steel pipe 11 is shaped by a buckling work to push it in the pipe axis direction from the outside with a punch member, a deep and large annular sharp wrinkle or pocket 17 is formed in the head inner circumference by the outward expansion accompanying the buckling work of the pushing operation. On the lower neck portion of the connecting head 12, on the other hand, there is fitted a washer (i.e., a connecting washer) 14 or a sleeve washer (i.e., a connecting washer) 15 either as a part necessary for applying an axial force effective for mounting the connecting head 12 on a mating seal face or as a reinforcement when vibrations occur.